In a parallel circuit, how is the equivalent resistance Rs determined?

• A. use Ohm's Law
• B. Rs = R1 + R2 + R3
• C. Rs = 1/(1/R1 + 1/R2 + 1/R3)
• D. Vs = V1 = V2 = V3

Answer: (A)

In a parallel circuit, the same voltage is applied to every branch and the total current is the sum of the branch currents. To find the equivalent resistance, treat the whole network as a single resistor connected to the source and use Ohm’s law. The equivalent resistance is the source voltage divided by the total current: Rs = Vs / It. If you know the branch resistances, each branch draws current Ii = Vs / Ri, so It = Σ(Vs / Ri) = Vs × Σ(1/Ri). Therefore Rs = Vs / It = 1 / Σ(1/Ri), which for three branches becomes Rs = 1/(1/R1 + 1/R2 + 1/R3). This shows why Ohm’s law is the appropriate tool to determine Rs in a parallel circuit. The fact that voltages across branches are equal is a property of parallel circuits, and adding resistances applies to series, not parallel.